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The mathematical discipline that studies the interdependence of the sides and angles of spherical triangles (see [[Spherical geometry|Spherical geometry]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867301.png" /> be the angles and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867302.png" /> be the opposite sides of a spherical triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867303.png" />. The angles and sides of the spherical triangle are related by the following basic formulas of spherical trigonometry:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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The mathematical discipline that studies the interdependence of the sides and angles of spherical triangles (see [[Spherical geometry|Spherical geometry]]). Let  $  A, B, C $
 +
be the angles and let  $  a, b, c $
 +
be the opposite sides of a spherical triangle  $  ABC $.  
 +
The angles and sides of the spherical triangle are related by the following basic formulas of spherical trigonometry:
 +
 
 +
$$ \tag{1 }
 +
 
 +
\frac{\sin  a }{\sin  A }
 +
  =
 +
\frac{\sin  b }{\sin  B }
 +
  =
 +
\frac{\sin  c }{\sin  C }
 +
 
 +
$$
  
 
(the sine theorem);
 
(the sine theorem);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867305.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\cos  a  = \cos  b  \cos  c + \sin  b  \sin  c  \cos  A
 +
$$
  
 
(the cosine theorem for sides);
 
(the cosine theorem for sides);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867306.png" /></td> </tr></table>
+
$$
 +
\cos  A  = - \cos  B  \cos  C + \sin  B  \sin  C  \cos  a
 +
$$
  
 
(the cosine theorem for angles);
 
(the cosine theorem for angles);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867307.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\sin  a  \cos  B  = \cos  b  \sin  c - \sin  b  \cos  c  \cos  A,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867308.png" /></td> </tr></table>
+
$$
 +
\sin  A  \cos  b  = \cos  B  \sin  C + \sin  B  \cos  C  \cos  a
 +
$$
  
(the formulas linking five elements). In these formulas, the sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s0867309.png" /> are measured by the corresponding central angles, and the lengths of these sides are equal respectively to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673013.png" /> is the radius of the sphere. By changing the notations of the angles and sides according to the circular permutation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673014.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673015.png" /> it is possible to write down other formulas of spherical trigonometry, analogous to those shown. The formulas of spherical trigonometry make it possible to determine any three elements of the spherical triangle from the other three.
+
(the formulas linking five elements). In these formulas, the sides $  a, b, c $
 +
are measured by the corresponding central angles, and the lengths of these sides are equal respectively to $  aR $,  
 +
$  bR $,  
 +
$  cR $,  
 +
where $  R $
 +
is the radius of the sphere. By changing the notations of the angles and sides according to the circular permutation: $  A \rightarrow B \rightarrow C \rightarrow A $
 +
$  ( a \rightarrow b \rightarrow c \rightarrow a) $
 +
it is possible to write down other formulas of spherical trigonometry, analogous to those shown. The formulas of spherical trigonometry make it possible to determine any three elements of the spherical triangle from the other three.
  
In order to find a spherical triangle by means of two given sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673016.png" /> and the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673017.png" /> between them, and by means of two given angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673018.png" /> and the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673019.png" /> between them, the following formulas are used (Napier analogues):
+
In order to find a spherical triangle by means of two given sides $  a, b $
 +
and the angle $  C $
 +
between them, and by means of two given angles $  A, B $
 +
and the side $  c $
 +
between them, the following formulas are used (Napier analogues):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\mathop{\rm tan}  A-  
 +
\frac{B}{2}
 +
  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673021.png" /></td> </tr></table>
+
\frac{\sin \{ ( a- b) / 2 \} }{\sin \{ ( a+ b) / 2 \} }
 +
  \mathop{\rm cot} 
 +
\frac{C}{2}
 +
,\ \
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$
 +
\mathop{\rm tan}  A+
 +
\frac{B}{2}
 +
  =
 +
\frac{\cos \{ ( a- b) / 2 \} }{
 +
\cos \{ ( a+ b) / 2 \} }
 +
  \mathop{\rm cot} 
 +
\frac{C}{2}
 +
;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673023.png" /></td> </tr></table>
+
$$ \tag{5 }
 +
\mathop{\rm tan}  a-
 +
\frac{b}{2}
 +
  =
 +
\frac{\sin \{ ( A- B) / 2 \} }{\sin \{ ( A+ B) / 2 \} }
 +
  \mathop{\rm tan} 
 +
\frac{c}{2}
 +
,\ \
 +
$$
  
For right-angled spherical triangles (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673025.png" /> is the hypotenuse, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673026.png" /> are the other two sides), the formulas are simplified, for example:
+
$$
 +
\mathop{\rm tan}  a+
 +
\frac{b}{2}
 +
  = 
 +
\frac{\cos \{ ( A- B) / 2 \} }{
 +
\cos \{ ( A+ B) / 2 \} }
 +
  \mathop{\rm tan} 
 +
\frac{c}{2}
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
+
For right-angled spherical triangles ( $  A = 90 \circ $,
 +
$  a $
 +
is the hypotenuse,  $  b, c $
 +
are the other two sides), the formulas are simplified, for example:
 +
 
 +
$$ \tag{1'}
 +
\sin  b  = \sin  a  \sin  B
 +
$$
  
 
(the sine theorem);
 
(the sine theorem);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2prm)</td></tr></table>
+
$$ \tag{2'}
 +
\cos  a  = \cos  b  \cos  c
 +
$$
  
 
(Pythagoras' spherical theorem);
 
(Pythagoras' spherical theorem);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3prm)</td></tr></table>
+
$$ \tag{3'}
 +
\sin  a  \cos  B  = \cos  b  \sin  c.
 +
$$
  
 
In solving problems, the following Delambre formulas, which link all six elements of a spherical triangle, are useful:
 
In solving problems, the following Delambre formulas, which link all six elements of a spherical triangle, are useful:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673030.png" /></td> </tr></table>
+
$$
 +
\sin 
 +
\frac{a}{2}
 +
  \cos  B-  
 +
\frac{C}{2}
 +
  = \sin 
 +
\frac{A}{2}
 +
  \sin  b+
 +
\frac{c}{2}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673031.png" /></td> </tr></table>
+
$$
 +
\sin 
 +
\frac{a}{2}
 +
  \sin  B-
 +
\frac{C}{2}
 +
  = \cos 
 +
\frac{A}{2}
 +
  \sin  b-  
 +
\frac{c}{2}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673032.png" /></td> </tr></table>
+
$$
 +
\cos 
 +
\frac{a}{2}
 +
  \cos  B+
 +
\frac{C}{2}
 +
  = \sin 
 +
\frac{A}{2}
 +
  \cos  b+
 +
\frac{c}{2}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673033.png" /></td> </tr></table>
+
$$
 +
\cos 
 +
\frac{a}{2}
 +
  \sin  B+
 +
\frac{C}{2}
 +
  = \cos 
 +
\frac{A}{2}
 +
  \cos  b-  
 +
\frac{c}{2}
 +
;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673034.png" /></td> </tr></table>
+
$$
 +
\sin 
 +
\frac{A}{2}
 +
  = \sqrt {
 +
\frac{\sin ( s- b)  \sin ( s- c) }{\sin  b  \sin  c }
 +
} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673035.png" /></td> </tr></table>
+
$$
 +
\cos 
 +
\frac{A}{2}
 +
  = \sqrt {
 +
\frac{\sin  s  \sin ( s- a)
 +
}{\sin  b  \sin  c }
 +
} ,\  s  = a+ b+
 +
\frac{c}{2}
 +
;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673036.png" /></td> </tr></table>
+
$$
 +
\sin 
 +
\frac{a}{2}
 +
  = \sqrt {
 +
\frac{- \cos  S  \cos ( S- A) }{\sin  B  \sin  C }
 +
} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673037.png" /></td> </tr></table>
+
$$
 +
\cos 
 +
\frac{a}{2}
 +
  = \sqrt {
 +
\frac{\cos ( S- B)  \cos ( S- C) }{\sin  B  \sin  C }
 +
} ,\  S  = A+ B+
 +
\frac{C}{2}
 +
.
 +
$$
  
 
For references, see [[Spherical geometry|Spherical geometry]].
 
For references, see [[Spherical geometry|Spherical geometry]].
 
 
  
 
====Comments====
 
====Comments====
 
The  "analogue"  in  "Napier analogues"  is an old-fashioned word for  "proportion" .
 
The  "analogue"  in  "Napier analogues"  is an old-fashioned word for  "proportion" .
  
From a relation between the elements of a spherical triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673038.png" /> with sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673039.png" />, a second relation can be derived by replacing every element by its supplement and at the same time replacing small letters by the corresponding capitals. An example:
+
From a relation between the elements of a spherical triangle $  ABC $
 +
with sides $  a, b, c $,
 +
a second relation can be derived by replacing every element by its supplement and at the same time replacing small letters by the corresponding capitals. An example:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673040.png" /></td> </tr></table>
+
$$
 +
\sin  a  \cos  B  = \cos  b  \sin  c - \sin  b  \cos  c  \cos  A
 +
$$
  
 
yields
 
yields
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673041.png" /></td> </tr></table>
+
$$
 +
\sin ( 180- A)  \cos ( 180- b) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673042.png" /></td> </tr></table>
+
$$
 +
= \
 +
\cos ( 180- B)  \sin ( 180- C) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673043.png" /></td> </tr></table>
+
$$
 +
- \sin ( 180- B)  \cos ( 180- C)  \cos ( 180- a) ,
 +
$$
  
 
i.e.
 
i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086730/s08673044.png" /></td> </tr></table>
+
$$
 +
\sin  A  \cos  b  = \cos  B  \sin  C +
 +
\sin  B  \cos  C  \cos  a .
 +
$$
  
 
Delambre's formulas are also called Gauss' formulas (or Gauss analogues).
 
Delambre's formulas are also called Gauss' formulas (or Gauss analogues).

Latest revision as of 08:22, 6 June 2020


The mathematical discipline that studies the interdependence of the sides and angles of spherical triangles (see Spherical geometry). Let $ A, B, C $ be the angles and let $ a, b, c $ be the opposite sides of a spherical triangle $ ABC $. The angles and sides of the spherical triangle are related by the following basic formulas of spherical trigonometry:

$$ \tag{1 } \frac{\sin a }{\sin A } = \frac{\sin b }{\sin B } = \frac{\sin c }{\sin C } $$

(the sine theorem);

$$ \tag{2 } \cos a = \cos b \cos c + \sin b \sin c \cos A $$

(the cosine theorem for sides);

$$ \cos A = - \cos B \cos C + \sin B \sin C \cos a $$

(the cosine theorem for angles);

$$ \tag{3 } \sin a \cos B = \cos b \sin c - \sin b \cos c \cos A, $$

$$ \sin A \cos b = \cos B \sin C + \sin B \cos C \cos a $$

(the formulas linking five elements). In these formulas, the sides $ a, b, c $ are measured by the corresponding central angles, and the lengths of these sides are equal respectively to $ aR $, $ bR $, $ cR $, where $ R $ is the radius of the sphere. By changing the notations of the angles and sides according to the circular permutation: $ A \rightarrow B \rightarrow C \rightarrow A $ $ ( a \rightarrow b \rightarrow c \rightarrow a) $ it is possible to write down other formulas of spherical trigonometry, analogous to those shown. The formulas of spherical trigonometry make it possible to determine any three elements of the spherical triangle from the other three.

In order to find a spherical triangle by means of two given sides $ a, b $ and the angle $ C $ between them, and by means of two given angles $ A, B $ and the side $ c $ between them, the following formulas are used (Napier analogues):

$$ \tag{4 } \mathop{\rm tan} A- \frac{B}{2} = \ \frac{\sin \{ ( a- b) / 2 \} }{\sin \{ ( a+ b) / 2 \} } \mathop{\rm cot} \frac{C}{2} ,\ \ $$

$$ \mathop{\rm tan} A+ \frac{B}{2} = \frac{\cos \{ ( a- b) / 2 \} }{ \cos \{ ( a+ b) / 2 \} } \mathop{\rm cot} \frac{C}{2} ; $$

$$ \tag{5 } \mathop{\rm tan} a- \frac{b}{2} = \frac{\sin \{ ( A- B) / 2 \} }{\sin \{ ( A+ B) / 2 \} } \mathop{\rm tan} \frac{c}{2} ,\ \ $$

$$ \mathop{\rm tan} a+ \frac{b}{2} = \frac{\cos \{ ( A- B) / 2 \} }{ \cos \{ ( A+ B) / 2 \} } \mathop{\rm tan} \frac{c}{2} . $$

For right-angled spherical triangles ( $ A = 90 \circ $, $ a $ is the hypotenuse, $ b, c $ are the other two sides), the formulas are simplified, for example:

$$ \tag{1'} \sin b = \sin a \sin B $$

(the sine theorem);

$$ \tag{2'} \cos a = \cos b \cos c $$

(Pythagoras' spherical theorem);

$$ \tag{3'} \sin a \cos B = \cos b \sin c. $$

In solving problems, the following Delambre formulas, which link all six elements of a spherical triangle, are useful:

$$ \sin \frac{a}{2} \cos B- \frac{C}{2} = \sin \frac{A}{2} \sin b+ \frac{c}{2} , $$

$$ \sin \frac{a}{2} \sin B- \frac{C}{2} = \cos \frac{A}{2} \sin b- \frac{c}{2} , $$

$$ \cos \frac{a}{2} \cos B+ \frac{C}{2} = \sin \frac{A}{2} \cos b+ \frac{c}{2} , $$

$$ \cos \frac{a}{2} \sin B+ \frac{C}{2} = \cos \frac{A}{2} \cos b- \frac{c}{2} ; $$

$$ \sin \frac{A}{2} = \sqrt { \frac{\sin ( s- b) \sin ( s- c) }{\sin b \sin c } } , $$

$$ \cos \frac{A}{2} = \sqrt { \frac{\sin s \sin ( s- a) }{\sin b \sin c } } ,\ s = a+ b+ \frac{c}{2} ; $$

$$ \sin \frac{a}{2} = \sqrt { \frac{- \cos S \cos ( S- A) }{\sin B \sin C } } , $$

$$ \cos \frac{a}{2} = \sqrt { \frac{\cos ( S- B) \cos ( S- C) }{\sin B \sin C } } ,\ S = A+ B+ \frac{C}{2} . $$

For references, see Spherical geometry.

Comments

The "analogue" in "Napier analogues" is an old-fashioned word for "proportion" .

From a relation between the elements of a spherical triangle $ ABC $ with sides $ a, b, c $, a second relation can be derived by replacing every element by its supplement and at the same time replacing small letters by the corresponding capitals. An example:

$$ \sin a \cos B = \cos b \sin c - \sin b \cos c \cos A $$

yields

$$ \sin ( 180- A) \cos ( 180- b) = $$

$$ = \ \cos ( 180- B) \sin ( 180- C) + $$

$$ - \sin ( 180- B) \cos ( 180- C) \cos ( 180- a) , $$

i.e.

$$ \sin A \cos b = \cos B \sin C + \sin B \cos C \cos a . $$

Delambre's formulas are also called Gauss' formulas (or Gauss analogues).

Figure: s086730a

References

[a1] H. Flanders, J. Price, "Trigonometry" , Acad. Press (1975)
[a2] G. Hessenberg, H. Kneser, "Ebene und Sphaerische Trigonometrie" , de Gruyter (1957)
[a3] W.A. Granville, P.F. Smith, J.S. Mikesh, "Spherical trigonometry" , Ginn (1943)
[a4] W. Lietzmann, "Elementare Kugelgeometrie" , Vandenhoeck & Ruprecht (1949)
[a5] M. Berger, "Geometry" , II , Springer (1989)
[a6] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
[a7] J.D.H. Donnay, "Spherical trigonometry after the Cesàro method" (1945)
How to Cite This Entry:
Spherical trigonometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_trigonometry&oldid=19309
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article